Exact solutions of (0,2) Landau-Ginzburg models 1608.07753 w/ A. Gadde Pavel Putrov IAS, Princeton November 28, 2016 1 / 17
Motivaiton Known: N p0, 0q: real boson φ, V pφq φ 2n IR ÝÑ n-th unitary minimal model SUp2q n 1ˆSUp2q 1 SUp2q n. N p2, 2q: chiral superfield Φ, W pφq Φ n`1 [Witten] IR ÝÑ n-th N 2 minimal model SUp2q n 1ˆSOp2q 1 Up1q. 2pn`1q Want: N p0, 2q LG??? IR ÝÑ (0,2) SCFT??? 2 / 17
(0,2) Landau-Ginzburg models Supercharges: Q Q Up1q rot `1{2 `1{2 Up1q R `1{2 1{2 tq, Qu 2P`, superspace: x`, x, θ`, θ`. Multiplets: Chiral: Φ b φ b ` ψ b`θ` `..., b 1... p Fermi: Ψ a ψ á `... `, a 1... q Interaction: ş dθ` řq a 1 Ψa J a pφq ù ř a J apφq 2 ` ř holomorphic a,b ψá BJ apφq Bφ b `c.c. ψ b` ` c.c. 3 / 17
Finding IR CFT UV data: tj a pφqu a 1...q ÝÑ IR CFT H à λ H L λ modules of VOA L, c L b HR λ modules of N 2 SVOA R, c R Tools/constraints to determine IR CFT: c-extrimization ñ c L, c R Superconformal index BPS spectrum (Ą Topological heterotic ring) t Hooft anomalies Modular invariance 4 / 17
Modular invariance Zpτq Tr H q L0 q L 0 ÿ λ χ L λ pqqχr λ p qq, pq e2πiτ q χ L λ pqq Tr Hλ L q L 0, χ R λ p qq Tr q L 0 HR λ Zpτ q should be modular invariant. NS sector: τ Ñ 1{τ, τ Ñ τ ` 2. anti-periodic 0 anti-periodic 1 Gives constraints on possible pairs (VOA L, SVOA R ). (e.g. pup1q 5, Up1q 3 q modular invariant pairing does not exist) 5 / 17
c-extremization [Benini-Bobev] Suppose vacuum is normalizable (tφ P C p J a pφq 0u is finite) pÿ qÿ c R 3 prrφ b s 1q 2 3 prrψ a sq 2 b 1 a 1 RrΦ b s, RrΨ a s trial R-charges. Extremize c R subject to conditions RrJ a pφqs ` RrΨ a s 1. c L c R q p (grav. anomaly) ñ c L... 6 / 17
t Hooft anomalies Up1q m flavor symmetry (similarly for non-abelian). k - m ˆ m anomaly matrix (integral quadratic form on charge lattice Z m ). A j J i xb µ J µ i y ř j k ijfµνɛ j µν k ij ÿ apfermi k is positive definite Ñ Up1q m k left-moving sector. q a i q a j ÿ bpchiral q b i q b j affine algebra in IR in the J i pzqj j p0q k ij z 2 vertex operators: e ř i ni ş J i, n P Z m. 7 / 17
BPS spectrum H BPS (IR) H L0` R 0 {2 0 à λ P chiral primaries H L λ H BPS (UV) Q-cohomology Maybe hard to calculate (though in principle possible). Simple finite-dimensional subsector: topological heterotic ring 8 / 17
Topological heterotic ring [Katz-Sharpe,Adam-Distler-Ernebjerg,Melnikov,...] Suppose there is a left-moving flavor symmetry Up1q L ü H L λ such that in UV: q L rφ b s RrΦ b s, q L rψ a s RrΨ a s 1. H Top (IR) H BPS L0 q L {2 H L0` R 0 {2 0, L 0 q L {2 Can be computed in the UV as Koszul homology: C 0 d ÝÑ ^qe d ÝÑ... d ÝÑ ^1E ÝÑ d ^0E E Span C t Ψ a u q a 1 b CrΦ is CrΦ i s q, d H Top (UV) H pc, dq Ker d{im d qÿ a 1 d ÝÑ 0 J a B B Ψ a, 9 / 17
Example 1 3 Φ 1 Φ 2 p 2, q 3 ż L int dθ` Ψ 1 Φ m 1 `Ψ 2 Φ n 2 `Ψ ` c.c. J 1 J 2 J 3 c-extr ñ c R 3 mn 1 mn ` 1, c L 2 2mn 1 mn ` 1 c R ă 3 ñ VOA R mn-th N 2 minimal model Up1q 2 flavor symmetry: q 1 q 2 Φ 1 1 0 Φ 2 0 1 Ψ 1 m 0 Ψ 2 0 n Ψ 3 1 1 k ˆ m2 1 1 n 2 ñ VOA L Ą Up1q 2 k, c L c Sugawara pup1q 2 k q 2 mn 2 mn ` 1 cpparafermionsq ă 2 10 / 17
Example 1, cont. IR CFT SUp2qmn 1 Up1q 2pmn 1q parafermions ˆUp1q 2 k b SUp2qmn 1 ˆ SOp2q 1 Up1q 2pmn`1q mn-th N 2 minimal model D modular invariant pairing: mn 1 ÿ Z α 0 checks: ÿ ν P Z 2pmn 1q ÿ Superconformal index Topological heterotic ring χ SUp2qmn 1{Up1q2pmn 1q α;ν a P Z mn`1 χ Up1q2 k pma, npa ` νqq PZ 2 {kz 2 χ SUp2q mn 1ˆSOp2q 1 {Up1q 2pmn`1q α;2a`ν 11 / 17
How to find modular invariant pairing (idea) cf. [Gannon] m Up1q 1 A Up1q m b 2 B m Up1q 3 C Up1q m 4 D D modular invariant pairing ô A D Q C B (as quadratic forms) Q-linear map ÞÑ pairing b (e.g. Up1q 5 b Up1q 3 : 5x 2 3y 2 3 ñ x y ñ no pairing) 5 x fy, f P HompQ m 2`m 4, Q m 1`m 3 q, s.t. pa Dqpxq pb Cqpyq. Mx gy, M P Z, g P HompZ m 2`m 4, Z m 1`m 3 q, s.t. pa Dqpgyq M 2 pb Cqpyq. ÿ µ 1,µ 2,µ 3,µ 4 χ Up1qm 1`m 3 A D pgyq ÿ pµ 1,µ 3 q ν 2,ν 4 A pµ1,µ 3 q,pν 2,ν 4 q χ Up1q m2`m 4 M 2 pb Cq pν 2,ν 4 q pyq, χ Up1qm 2`m 4 B C pmyq ÿ Up1q m2`m 4 M B pµ 2,µ 4 q pµ2,µ 4 q,pν 2,ν 4 q χ 2 pb Cq pyq. pν 2,ν 4 q ν 2,ν 4 ÿ A pµ1,µ 3 q,pν 2,ν 4 q B pµ2,µ 4 q,pν 2,ν 4 q χ Up1q m 1 A {Up1qm 2 B µ 1 ;µ 2 χ Up1qm 3 C {Up1qm 4 D µ 3 ;µ 4 ν 2,ν 4 is modular invariant. 12 / 17
Example 1, cont. Check that indeed UV and IR calculations of the topological heterotic ring agree: H L0` R 0 {2 0, L 0 q L {2 H pc, dq H 3 pc, dq 0, H 2 pc, dq 0, H 1 pc, dq Span C t Ψ 3 Φ m 1 1 Φ b 2 Ψ 1 Φ b`1 2 u n 1 b 0 Span C t Ψ 3 Φ n 1 2 Φ a 1 Ψ 2 Φ a`1 1 u m 2 a 0, H 0 pc, dq Span C tφ aum 1 1 a 0 Span C tφbun 1 2 b 1, q 2 (m,n)=(5,4) 4 3 2 1 H 0 (C,d) H 1 (C,d) 0 1 2 3 4 5 q 1 13 / 17
Example 2 IR CFT p 2, q 2 ż L int dθ` Ψ 1 pφ m 1 ` Φ n 2 q `Ψ 2 Φ 1 Φ 2 ` c.c. J 1 J 2 SUp2qmn Up1q 2mn parafermions ˆUp1q mnpmn`2q b SUp2qmn ˆ SOp2q 1 Up1q 2pmn`2q mn ` 1-th N 2 minimal model Modular invariant pairing depends on pm, nq individually (inequivalent Q-linear maps px 1, x 2 q ÞÑ py 1, y 2 q such that mnpmn ` 2q x 2 1 ` 2pmn ` 2q x2 2 2mn y2 1 ` y2 2 ) 14 / 17
Example 2, cont. mn 6: 12x 2 1 ` x2 2 Q 48y 2 1 ` 16y2 2 (m,n)=(1,6) ˆ x1 x 2 ˆ 1 1 6 2 ˆ y1 y 2 6à à H λ 0 αpz 12 à H SUp2q6{Up1q12 λ;ᾱ bh Up1q 48 6s`α b H SUp2q 6ˆSOp2q 1 {Up1q 16 λ;2s α spz 8 (m,n)=(2,3) ˆ x1 x 2 1 7 ˆ 11 5 30 22 ˆ y1 y 2 6à à H λ 0 αpz 12 à H SUp2q6{Up1q12 λ;ᾱ bh Up1q 48 6s`5α b H SUp2q 6ˆSOp2q 1 {Up1q 16 λ;2s 5α spz 8 15 / 17
Example 3 ż L int dθ` Nÿ i,j 1 Ψ ij Φ i Φ j Nonabelian flavor symmetry: UpNq SUpNqˆUp1q Z N. IR CFT `SUpNq N`1 ˆ Up1q Np2N`1q b SOp2N ` 2q1 ˆ SOpNpN ` 1qq 1 SUpN ` 1q N ˆ Up1q pn`1qp2n`1q Kazama-Suzuki coset rsop2n ` 2q{UpN ` 1qs 2N`1 Modular invariant pairing provided by level-rank duality: conformal Up1q 1 ˆ UpNpN ` 1qq 1 Ą `SUpNqN`1 ˆ Up1q Np2N`1q ˆ `SUpN ` 1qN ˆ Up1q pn`1qp2n`1q 16 / 17
Summary For a number of families of N p0, 2q LG models we found explicit descritption of the N p0, 2q SCFT at the IR fixed point in terms of (KS) WZW cosets. To determine the IR CFT we use RG protected quantities (BPS spectrum and anomalies) as well as modular invariance In particular we give explicit expression of the full partition function of the IR CFT (which is not a supersymetric partition function, it could not be computed by localization) 17 / 17